Partial inverse problems for quadratic differential pencils on a graph with a loop

  • 1 Department of Applied Mathematics and Physics, Samara National Research University, Moskovskoye Shosse 34, Samara, Russia
  • 2 Department of Mathematics, Tamkang University, 151 Ying-chuan Road Tamsui, New Taipei, Taiwan
Natalia P. BondarenkoORCID iD: https://orcid.org/0000-0003-2513-1472
  • Corresponding author
  • Department of Applied Mathematics and Physics, Samara National Research University, Moskovskoye Shosse 34, Samara, 443086, Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia
  • orcid.org/0000-0003-2513-1472
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and Chung-Tsun ShiehORCID iD: https://orcid.org/0000-0003-1985-173X

Abstract

In this paper, partial inverse problems for the quadratic pencil of Sturm–Liouville operators on a graph with a loop are studied. These problems consist in recovering the pencil coefficients on one edge of the graph (a boundary edge or the loop) from spectral characteristics, while the coefficients on the other edges are known a priori. We obtain uniqueness theorems and constructive solutions for partial inverse problems.

  • [1]

    G. Berkolaiko, R. Carlson, S. A. Fulling and P. Kuchment, Quantum Graphs and Their Applications, Contemp. Math. 415, American Mathematical Society, Providence, 2006.

  • [2]

    N. Bondarenko and S. Buterin, On recovering the Dirac operator with an integral delay from the spectrum, Results Math. 71 (2017), no. 3–4, 1521–1529.

    • Crossref
    • Export Citation
  • [3]

    N. Bondarenko and C.-T. Shieh, Partial inverse problems on trees, Proc. Roy. Soc. Edinburgh Sect. A 147 (2017), 917–933.

    • Crossref
    • Export Citation
  • [4]

    N. P. Bondarenko, A partial inverse problem for the differential pencil on a star-shaped graph, Results Math. 72 (2017), no. 4, 1933–1942.

    • Crossref
    • Export Citation
  • [5]

    N. P. Bondarenko, A 2-edge partial inverse problem for the Sturm–Liouville operators with singular potentials on a star-shaped graph, Tamkang J. Math. 49 (2018), no. 1, 49–66.

    • Crossref
    • Export Citation
  • [6]

    N. P. Bondarenko, A partial inverse problem for the Sturm–Liouville operator on a star-shaped graph, Anal. Math. Phys. 8 (2018), no. 1, 155–168.

    • Crossref
    • Export Citation
  • [7]

    N. P. Bondarenko, Partial inverse problems for the Sturm–Liouville operator on a star-shaped graph with mixed boundary conditions, J. Inverse Ill-Posed Probl. 26 (2018), no. 1, 1–12.

    • Crossref
    • Export Citation
  • [8]

    N. P. Bondarenko, Inverse problem for the differential pencil on an arbitrary graph with partial information given on the coefficients, Anal. Math. Phys. 9 (2019), no. 3, 1393–1409.

    • Crossref
    • Export Citation
  • [9]

    N. P. Bondarenko and C.-F. Yang, Partial inverse problems for the Sturm–Liouville operator on a star-shaped graph with different edge lengths, Results Math. 73 (2018), no. 2, Paper No. 56.

  • [10]

    S. A. Buterin, G. Freiling and V. A. Yurko, Lectures in the theory of entire functions, Schriftenreihe der Fakultät für Matematik SM-UDE-779, Duisbug-Essen University, 2014.

  • [11]

    S. A. Buterin and V. A. Yurko, Inverse spectral problem for pencils of differential operators (in Russian), Vestnik Bashkir. Univ. (2006), no. 4, 8–12.

  • [12]

    S. A. Buterin and V. A. Yurko, Inverse problems for second-order differential pencils with Dirichlet boundary conditions, J. Inverse Ill-Posed Probl. 20 (2012), no. 5–6, 855–881.

  • [13]

    P. Exner, J. P. Keating, P. Kuchment, T. Sunada and A. Teplyaev, Vladimir A. Geyler, April 29, 1943–April 2, 2007, Analysis on Graphs and its Applications, Proc. Sympos. Pure Math. 77, American Mathematical Society, Providence (2008), 1–8.

  • [14]

    G. Freiling and V. Yurko, Inverse Sturm–Liouville Problems and Their Applications, Nova Science, Huntington, 2001.

  • [15]

    M. G. Gasymov and G. v. Guseĭnov, Determination of a diffusion operator from spectral data, Akad. Nauk Azerbaĭdzhan. SSR Dokl. 37 (1981), no. 2, 19–23.

  • [16]

    F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential. II. The case of discrete spectrum, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2765–2787.

    • Crossref
    • Export Citation
  • [17]

    H. Hochstadt and B. Lieberman, An inverse Sturm–Liouville problem with mixed given data, SIAM J. Appl. Math. 34 (1978), no. 4, 676–680.

    • Crossref
    • Export Citation
  • [18]

    R. Hryniv and N. Pronska, Inverse spectral problems for energy-dependent Sturm–Liouville equations, Inverse Problems 28 (2012), no. 8, Article ID 085008.

  • [19]

    R. O. Hryniv and Y. V. Mykytyuk, Half-inverse spectral problems for Sturm–Liouville operators with singular potentials, Inverse Problems 20 (2004), no. 5, 1423–1444.

    • Crossref
    • Export Citation
  • [20]

    B. J. Levin and J. I. Ljubarskiĭ, Interpolation by entire functions belonging to special classes and related expansions in series of exponentials, Math. USSR-Izv. 9 (1975), no. 3, 621–662.

    • Crossref
    • Export Citation
  • [21]

    O. Martinyuk and V. Pivovarchik, On the Hochstadt–Lieberman theorem, Inverse Problems 26 (2010), no. 3, Article ID 035011.

  • [22]

    N. Pronska, Reconstruction of energy-dependent Sturm–Liouville equations from two spectra, Integral Equations Operator Theory 76 (2013), no. 3, 403–419.

    • Crossref
    • Export Citation
  • [23]

    N. I. Pronska, Asymptotics of eigenvalues and eigenfunctions of energy-dependent Sturm–Liouville equations, Mat. Stud. 40 (2013), no. 1, 38–52.

  • [24]

    L. Sakhnovich, Half-inverse problems on the finite interval, Inverse Problems 17 (2001), no. 3, 527–532.

    • Crossref
    • Export Citation
  • [25]

    C.-F. Yang, Inverse spectral problems for the Sturm–Liouville operator on a d-star graph, J. Math. Anal. Appl. 365 (2010), no. 2, 742–749.

    • Crossref
    • Export Citation
  • [26]

    C.-F. Yang and F. Wang, Inverse problems on a graph with loops, J. Inverse Ill-Posed Probl. 25 (2017), no. 3, 373–380.

  • [27]

    C.-F. Yang and X.-P. Yang, Uniqueness theorems from partial information of the potential on a graph, J. Inverse Ill-Posed Probl. 19 (2011), no. 4–5, 631–641.

  • [28]

    V. Yurko, Recovering differential pencils on compact graphs, J. Differential Equations 244 (2008), no. 2, 431–443.

    • Crossref
    • Export Citation
  • [29]

    V. Yurko, Inverse problems for non-selfadjoint quasi-periodic differential pencils, Anal. Math. Phys. 2 (2012), no. 3, 215–230.

    • Crossref
    • Export Citation
  • [30]

    V. Yurko, Inverse spectral problems for differential pencils on a graph with a rooted cycle, Inverse Probl. Sci. Eng. 24 (2016), no. 9, 1647–1660.

    • Crossref
    • Export Citation
  • [31]

    V. Yurko, Inverse problems for differential pencils on A-graphs, J. Inverse Ill-Posed Probl. 25 (2017), no. 6, 819–828.

  • [32]

    V. Yurko, Inverse problems for differential pencils on bush-type graphs, Results Math. 71 (2017), no. 3–4, 1047–1062.

    • Crossref
    • Export Citation
  • [33]

    V. Yurko, Inverse spectral problems for differential pencils on arbitrary compact graphs, Differ. Equ. 55 (2019), no. 1, 24–33.

    • Crossref
    • Export Citation
  • [34]

    V. A. Yurko, An inverse problem for differential pencils on graphs with a cycle, J. Inverse Ill-Posed Probl. 22 (2014), no. 5, 625–641.

  • [35]

    V. A. Yurko, Inverse spectral problems for differential operators on spatial networks, Russian Math. Surveys 71 (2016), no. 3(429), 539–584.

    • Crossref
    • Export Citation
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